Optical fiber, optical fiber filter, and optical amplifier

ABSTRACT

An optical fiber filter includes an optical fiber, a pair of first coupling regions, and a phase-shift region. The optical fiber includes a core and a cladding. An optical signal can pass through the optical fiber. The first coupling regions are formed in the optical fiber, at a predefined mutual distance, for producing power transfer between first and second propagation modes of the optical signal. The phase-shift region is defined by a section of the optical fiber, disposed between the first coupling regions, for producing a phase shift between the first and second propagation modes of the optical signal. In the first coupling regions, the optical fiber includes, in cross-section, an asymmetrical refractive index profile. A related optical fiber filtering device, optical fiber, process for producing an optical filter, optical amplifier, optical telecommunications system, optical fiber modal coupler, and method for filtering an optical signal are also disclosed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a national stage entry under 35 U.S.C. § 371 fromInternational Application No. PCT/EP01/12222, filed Oct. 23, 2001, inthe European Patent Office, the contents of which are relied upon andincorporated herein by reference; additionally, Applicants claim theright of priority under 35 U.S.C. § 119(a)–(d) based on patentapplication No. 00123718.9, filed Oct. 31, 2000, in the European PatentOffice; further, Applicants claim the benefit under 35 U.S.C. § 119(e)based on prior-filed, copending provisional application No. 60/246,069,filed Nov. 7, 2000, in the U.S. Patent and Trademark Office.

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention relates to an optical filtering device made froman optical fibre, in particular an optical fibre filter which can beused in a system for the transmission of wavelength divisionmultiplexing (abbreviated “WDM”) optical signals. The present inventionalso relates to a process for the manufacture of this filter, to anoptical fibre which can be used to form this filter, to a system for thetransmission of WDM signals using this filter and to a method forfiltering optical signals.

Description of the Related Art

In detail, a WDM optical signal is a (digital or analogic) signalcomprising a plurality of N optical signals which are independent ofeach other and each of which has a respective transmission wavelengthλ₁, λ₂ . . . λ_(N) different from that of the other signals. Eachtransmission wavelength defines a transmission “channel”. Moreover, eachsignal has, associated with it, a respective wavelength bandwidth Δλ ofpredefined size—called a “channel bandwidth” or “channel (spectral)size”—which is centred on the corresponding transmission wavelength. Thechannel size depends, typically, on the characteristics of the lasersources used and on the type of modulation used in order to associatethe information to be transmitted with the signal. In the absence ofmodulation, typical spectral amplitude values of a signal emitted by alaser source are in the region of 10 MHz, while in the case ofmodulation outside the range of 2.5 Gbit/s they are in the region of 5GHz.

The WDM signal also has a “spacing between channels” defined as thewavelength (or—in an equivalent manner—frequency) separation between thecentral wavelengths of two adjacent channels. In order to transmit to ahigh number of channels in one of the so-called “transmission windows”of the optical fibres and in a useful amplification bandwidth of theoptical amplifiers, the spacing between the channels of a WDM signal is,typically, in the region of one nanometre.

Generally, in a WDM system, the transmission of signals occurs in thefollowing manner: the various signals are first generated by respectiveoptical sources, then multiplexed so as to form a WDM signal, nexttransmitted along the same optical fibre transmission line and, finally,demodulated so as to be received by respective receivers.

In recent wavelength multiplexing optical amplification and transmissionsystems (able to transmit, along the same fibre, a very high number ofchannels—for example 128—distributed over a particularly wide spectralbandwidth—for example 70 nm) and, more generally, in optical signalprocessing apparatus, for both instrumentation and sensors, devices madeentirely of optical fibre, without any propagation of the light in freespace, are being increasingly used. In particular, these devices arerequired for the operations of spectral filtering, multiplexing anddemultiplexing of the channels and separation of the transmissionspectrum into bands.

With regard to spectral filtering, it is necessary to use both deviceswith a high wavelength selectivity for filtering of individual channelsand wider-bandwidth devices for equalization of the channels in theamplification bandwidth of the optical amplifiers. Equalization of thechannels is necessary since the gain spectrum of erbium-doped opticalfibre amplifiers (which constitute the most widely used opticalamplification means) has a significantly unequalized form in the regionbetween 1530 and 1560 nm. Despite the progress achieved in thedevelopment of glass matrices for silica-based optical fibres containingvarious co-dopants able to “flatten” the spectral gain curve, at themoment silica-based fibres which have a sufficiently uniform gainprofile such as not to require external equalization are not available.

The configuration most used to form wideband optical gain modulesinvolves the use of an equalizer filter arranged between twoactive-fibre amplification stages. The insertion of the filter betweenthe two amplification stages has the fundamental advantage of allowing aspectral “redistribution” of the power available for amplification,instead of simple suppression of the power in the wavelength regionswith a higher gain. The spectral profile of the filter which offersmaximum equalization depends on the operating conditions of theamplifier (and, therefore, on the power of the pump radiation suppliedto each stage) as well as the number and the wavelength distribution ofthe channels. In recent systems where there is the possibility ofchannel addition/extraction, the number and the distribution of thechannels may change depending on the configuration chosen by the systemmanager.

For the abovementioned reasons it has become important to have opticalfilters which can be efficiently integrated with the active fibre, withlow insertion losses and with a spectral profile which can be easilymodified depending on the specific use of the individual amplifier.

Different types of filters made directly using optical fibre are known.

A first type of filter is distinguished by the fact that the fibre has aportion with a sudden variation in diameter, i.e. a tapered portion.This region induces, in each signal passing through it, an attenuationwhich depends on the wavelength of the said signal. In this way,therefore, spectral filtering is performed. The spectral form of theattenuation of these filters is substantially sinusoidal on thewavelength.

Another type of filter, which is known as a “Fabry-Perot” filter, isformed by an optical fibre and two Bragg gratings formed in the fibreitself and operating as mirrors so as to define an optical resonator.

More recently so-called long period gratings (LPG) have been developed,said gratings being distinguished by periodic variations in the indexprofile of a fibre (typically by means of exposure to UV radiation) andalso allowing wavelength filtering to be performed.

A further class of filters is that defined by an interferometricstructure of the Mach-Zehnder type. Such a structure must be able toperform separation of an optical signal into two different distributionsof electromagnetic field, propagate these distributions along respectiveoptical paths into which it is possible to introduce, in a controlledmanner, a mutual delay and, subsequently, combine again the twoelectromagnetic field distributions so as to obtain an opticalinterference signal, the intensity of which is a function of thewavelength.

FIG. 1 shows schematically a Mach-Zehnder filter 50 of known type, ableto operate with two distinct field distributions. This interferometricstructure comprises a first and a second optical fibre 51, 52 joined attwo different points by means of a first and a second fusion coupler 53,54, for example of the 50/50 (or 3 dB) type. The filter 50 is able toreceive at its input a signal S_(in) from a first end of the first fibre51 and provide at its output a filtered signal S_(out) to a second endof the first fibre 51. In the section between the couplers 53, 54, thefibres 51, 52 define optical paths of different length. The differencein optical path length between the two fibres 51, 52 may be due to thefact that they have different transmissive properties, so that thesignals which are propagated in one fibre have a different speed fromthose which are propagated in the other fibre or, as shown in thefigure, may be due to the fact that they have different lengths L andL+ΔL in the section considered.

The couplers 53, 54 allow power coupling between the electromagneticfields which are propagated in the two fibres 51, 52. In particular, thefunction of the first coupler 53 is that of exciting two differentelectromagnetic field distributions in the optical fibres 51, 52 fromthe signal S_(in). These electromagnetic fields, which are propagatedalong different optical paths, accumulate a relative phase difference Δφwhich is not zero and defined by:

$\begin{matrix}{{{\Delta\phi}(\lambda)} = \frac{2{\pi \cdot n_{eff} \cdot \Delta}\; L}{\lambda}} & (1)\end{matrix}$

where n_(eff) is the effective refractive index of the mode which ispropagated in the fibres, λ is the wavelength and ΔL is the differencein length between the sections of the two fibres 51, 52 comprisedbetween the two couplers 53, 54.

The second coupler 54 is designed to combine again the twoelectromagnetic fields, generating an interference between them whichmay be constructive or destructive, depending on the phase shift Δφaccumulated.

In the simplest case where the fibres 51, 52 are identical and thecouplers 53, 54 have an optical power dividing ratio equal to 50/50 (3dB couplers), the optical powers at the two outputs of the secondcoupler 54, indicated respectively by P₁ and P₂, are defined by thefollowing equations:

$\begin{matrix}\begin{matrix}{P_{1} = {\cos^{2}\left( \frac{{\Pi \cdot n_{eff} \cdot \Delta}\; L}{\lambda} \right)}} \\{P_{2} = {\sin^{2}\left( \frac{{\Pi \cdot n_{eff} \cdot \Delta}\; L}{\lambda} \right)}}\end{matrix} & (2)\end{matrix}$

FIG. 2 shows the normalised transmission spectrum T(λ) of the filter 50at the output of its first fibre 51, in the case where ΔL is equal to 5μm and n_(eff) is equal to 1.462. The period of this curve is notconstant and is a function of the characteristics of the waveguidesused. Having a different response for the different wavelengths, theinterferometer may be advantageously used as an optical filter.

A Mach-Zehnder filter such as that described above is, however,difficult to use in practice, owing to its extreme sensitivity toexternal disturbances (for example variations in temperature) andvariations in form (in particular variations in curvature of the fibre).These phenomena cause variations in the effective refractive indexn_(eff) and, therefore, in the optical path, which are generallydifferent for the two fibres. The behaviour of this device, which isideally described by the equations (1) and (2), therefore cannot bepredicted precisely in a real situation.

In order to overcome this drawback, a solution which combines the twowaveguides in a single compact structure has been proposed. The U.S.Pat. No. 5,295,205 in the name of Corning proposes a filter formed byintroducing two optical fibres which are different from each otherinside a glass tube, collapsing the tube onto the fibres after creatinga vacuum inside the tube and, finally, heating and stretching the tubein two regions located at a distance from each other so as to form twotapered regions which define modal couplers. The fibres also havedifferent propagation constants in the zone lying between the twocouplers, resulting in a relative delay between the optical signalspropagated therein.

The Applicant considers that this solution is difficult to realise onaccount of the technological complexity of certain steps in theproduction process, in particular the operations for collapsing theglass tube around the fibres after creating a vacuum in the tube andforming the couplers at a distance from each other determined on thebasis of the desired spectral form and independently of the geometry ofthe tapered region.

An alternative method of producing a Mach-Zehnder interferometer is thatdescribed in international patent application WO00/00860 in the name ofCorning. This document describes a coaxial optical device comprising anoptical fibre and a coupling regulator integral with the optical fibre.The optical fibre is single-mode in the third spectral window of opticaltelecommunications and a glass tube with a refractive index lower thanthat of the cladding is collapsed onto the fibre, as described in thealready mentioned U.S. Pat. No. 5,295,205. In the region where thecollapsed tube is present, the refractive index profile is modified soas to allow locally the transmission of two modes, in particular themodes LP₀₁ and LP₀₂. These modes, which are mutually perpendicular bydefinition, define two distinct field distributions which, as they arepropagated, accumulate a relative phase difference Δφ. In the regionoccupied by the glass tube, non-adiabatic tapered zones able to inducepower coupling between the modes are formed. The tapered zones areformed by means of the normal technique for manufacturing fusioncouplers, by causing sudden reductions in the diameter of the fibre andthe tube collapsed onto it, such as to obtain coupling between thesymmetrical modes LP₀₁ and LP₀₂, but avoid coupling with the mode LP₀₃.

The Applicant also notes that the device described above requires theexecution of technologically complex manufacturing steps, such ascollapsing of a glass tube, under vacuum, onto an optical fibre and theformation of non-adiabatic tapered zones such as to have a high value ofthe coupling factor between the symmetrical modes LP₀₁ and LP₀₂, butwithout exciting other higher symmetrical modes such as the mode LP₀₃(where “coupling factor” or “splitting ratio” is understood in this caseas being the ratio between the power transferred to the mode LP₀₂ andthe remaining power in the mode LP₀₁).

The Applicant therefore notes that the Mach-Zehnder optical fibrefilters of the known type are made using complex technology which doesnot allow easy control of the filter parameters. The critical nature ofthe manufacturing process therefore results in high costs and fairly lowproduction outputs.

SUMMARY OF THE INVENTION

The Applicant has considered the problem of providing a Mach-Zehnderoptical fibre filter which is easy to produce, compact and has a highperformance.

The Applicant has found that a Mach-Zehnder interferometer which is easyand inexpensive to manufacture and has predetermined spectralcharacteristics may be made using a dual-mode fibre designed to allowpropagation of the fundamental mode LP₀₁ and the asymmetrical mode LP₁₁and provided with two modal coupling regions (for coupling the modesLP₀₁ and LP₁₁) in which the refractive index profile is asymmetrical dueto the presence of a cladding zone with a higher refractive index. Thiszone defines essentially, viewed in cross section, an annular sector ofthe cladding in a region adjacent to the core and has a radial extensioncorresponding substantially to that of the mode LP₁₁.

The Applicant has found that a filter with coupling regions of this typemay be made using an optical fibre having the innermost region of thecladding doped so as to provide it with high thermo-refractiveproperties and by thermally stressing this region so as to produce thedesired asymmetrical and localised variation in the index profile. Thisdoping may be performed with germanium, phosphorus and fluorine and mustbe such that the fibre is able to respond to a thermal stress ofsuitable intensity with a variation in the refractive index greater than5·10⁻⁴, preferably greater than or equal to 10⁻³, more preferablygreater than or equal to 2·10⁻³.

The Applicant has also found that the thermal stressing may be performedby means of the electric arc of a fusion jointer. The Applicant hasfound that this technique is particularly simple and flexible and may beused to produce very localised disturbances in the cross section of theoptical fibre.

According to a first aspect, the present invention relates to an opticalfibre filter comprising:

-   -   an optical fibre which includes a core and a cladding and        through which an optical signal can pass;    -   a pair of coupling regions formed in said optical fibre at a        predefined mutual distance, for producing a power transfer        between a first and a second propagation mode of said optical        signal;    -   a phase shift region, defined by a section of said fibre lying        between said coupling regions, for producing a phase shift        between said first and said second propagation modes;

in which, in said coupling regions, said optical fibre has, in crosssection, an asymmetrical refractive index profile.

Preferably, in each of said coupling regions, said cladding has, incross section, an annular sector in which the refractive index isgreater than that of the remainder of said cladding.

The cladding has, in cross section, an inner annular region adjacent tothe core and an outer annular region, said annular sector preferablybelonging to said inner annular region.

The annular sectors of said coupling regions preferably havesubstantially the same angular position.

The inner annular region has an internal radius r₁ and an externalradius r₂=k·r₁, in which k is preferably between 2 and 6.

Outside of said coupling regions, said optical fibre has, in crosssection, a refractive index profile preferably of the step index type.

Said optical signal has a wavelength comprised in a predefinedtransmission wavelength band and said optical fibre is preferablydual-mode in said wavelength band. Moreover, the filter comprises afirst and a second optical connection fibres which are single-mode insaid wavelength band and connected to opposite ends of said opticalfibre.

Advantageously, the inner annular region comprises silica and oxides ofthe following elements: germanium, phosphorus and fluorine.

The filter preferably comprises a further pair of coupling regionsformed in said optical fibre, in each of which said cladding has, incross section, a further annular sector in which the refractive index isgreater than that of the remainder of said cladding, said furtherannular sectors having substantially the same angular position,different from that of said angular sectors.

Preferably, each coupling region of said further pair of couplingregions is formed in the vicinity of a respective coupling region ofsaid pair of coupling regions.

Advantageously, the difference between the refractive index in saidannular sector and the refractive index of the remainder of saidcladding is equal to at least 5·10⁻⁴ and, more preferably, is equal toat least 2·10⁻³.

The filter according to the present invention may comprise a pluralityof filters as defined above, connected in series.

According to a further aspect, the present invention relates to anoptical fibre which can be used for producing a filter as defined above,comprising a core and a cladding, the cladding having a radially innerregion adjacent to the core and a radially outer region, in which saidradially inner region has a composition such as to obtain a variation inrefractive index equal to at least 5·10⁻⁴ following thermal stressingand in which said optical fibre is dual-mode in a wavelength band lyingbetween 1500 nm and 1650 nm. Preferably, said variation in refractiveindex is equal to at least 1·10⁻³ and, more preferably, is equal to atleast 2·10⁻³.

The difference n₂−n₃ between the refractive index n₂ in said radiallyinner region and the refractive index n₃ in said radially outer regionis preferably between +1·10⁻³ and −2·10⁻³.

Said radially inner region preferably comprises silica and oxides of thefollowing elements: germanium, phosphorus and fluorine. Advantageously,in said radially inner region, the germanium has a concentration ofbetween 2% and 5%, the phosphorus has a concentration of between 0.5%and 2% and the fluorine has a concentration of between 1% and 2%.

Preferably, the core comprises silica and at least one element selectedfrom germanium and phosphorus.

Preferably, the fibre is dual-mode in a wavelength band of between 1500nm and 1650 nm.

Said inner annular region has an internal radius r₁ and an externalradius r₂=k·r₁, in which k is preferably between 2 and 6.

According to another aspect, the present invention relates to a processfor the production of an optical filter from an optical fibre as definedabove, comprising the step of applying an electric arc to a first and asecond portions of said optical fibre in such a way as to stress thecladding of said optical fibre thermally in an asymmetrical manner.

Advantageously, said electric arc is generated between a pair ofelectrodes and the process comprises the step of displacing said opticalfibre in a controlled manner relative to said electrodes after applyingthe electric arc to said first portion and before applying the electricarc to said second portion.

Said electric arc has a duration preferably less than 400 ms, and morepreferably less than 300 ms, and has a current intensity preferably ofbetween 8 and 14 mA and more preferably between 10 and 11 mA.

In order to disturb thermally said first portion and said secondportion, instead of applying a single arc, a plurality of electric arcsmay be applied in succession.

According to a further aspect, the present invention relates to anoptical amplifier comprising at least one optical amplification stageand an optical filter as defined above, arranged in series with saidoptical amplification stage.

According to a further aspect, the present invention relates to anoptical telecommunications system comprising at least one opticaltransmitter, at least one optical receiver, an optical transmission lineconnecting said transmitter to said receiver and at least one opticalamplifier arranged along said transmission line, in which said opticalamplifier comprises at least one optical amplification stage and anoptical filter as defined above, arranged in series with said opticalamplification stage.

Preferably, said optical amplifier comprises two optical amplificationstages and said optical filter is arranged between said two stages.Alternatively, said optical filter is arranged downstream of said twostages.

According to a further aspect, the present invention relates to anoptical-fibre modal coupler comprising:

-   -   an optical fibre which comprises a core and a cladding and        through which an optical signal can pass; and    -   a coupling region formed in said optical fibre for producing a        power transfer between a first and a second propagation mode of        said optical signal;

in which said optical fibre has, in cross section, an asymmetricalrefractive index profile in said coupling region.

According to a further aspect, the present invention relates to a methodfor filtering an optical signal, said optical signal being transmittedin a waveguide in the fundamental mode LP₀₁, the method comprising thesteps of:

-   -   transmitting said signal through a first waveguide region        having, in cross section, an asymmetrical refractive index        profile so as to transfer power from the fundamental mode LP₀₁        to the asymmetrical mode LP₁₁;    -   conveying said fundamental mode LP₀₁ and said asymmetrical mode        LP₁₁ over a predefined distance so as to produce a relative        phase shift depending on said distance and the wavelength;    -   transmitting said fundamental mode LP₀₁ and said asymmetrical        mode LP₁₁ through a second waveguide region having, in cross        section, an asymmetrical refractive index profile, so as to        couple power between the fundamental mode LP₀₁ and the        asymmetrical mode LP₁₁.

BRIEF DESCRIPTION OF THE DRAWINGS

Further details may be obtained from the following description whichrefers to the accompanying figures listed below:

FIG. 1 shows in schematic form a Mach-Zehnder filter of known type;

FIG. 2 shows the transmission spectrum of the filter according to FIG.1;

FIG. 3 a shows a schematic and partial view of a Mach-Zehnder filterproduced in accordance with the invention;

FIG. 3 b shows a cross section through line III—III of the filter ofFIG. 3 a;

FIG. 3 c shows an overall schematic view of the filter according to theinvention;

FIG. 4 shows the refractive index profile of an optical fibre which canbe used in order to produce the filter according to FIG. 3;

FIG. 5 shows in schematic form an apparatus for forming the modalcoupling regions of the filter according to the invention;

FIG. 6 shows in schematic form a step in the process for the productionof the filter according to FIG. 3 a, in which a predefined section ofoptical fibre is struck by the electric arc of a fusion jointer;

FIGS. 7 a and 7 b show respectively the refractive index profile of thefibre of the filter according to the invention in a section thermallydisturbed by the process step according to FIG. 6 and in a section notdisturbed thermally;

FIG. 8 shows an apparatus which can be used for monitoring the couplingcharacteristics of the filter during formation of the modal couplingregions;

FIG. 9 shows the transmission spectrum, obtained by means of theapparatus according to FIG. 8, of a filter produced in accordance withthe invention;

FIG. 10 shows schematically an experimental apparatus for measuring themodal coupling due to the asymmetrical variation in the refractive indexprofile of the fibre;

FIGS. 11 a and 11 b show the results of a measurement carried out withthe apparatus according to FIG. 10, following a regression (fitting)operation;

FIG. 12 shows a measurement of the losses of the filter according to theinvention, due to polarisation of the input signal;

FIG. 13 shows a different embodiment of the filter according to theinvention;

FIGS. 14 a and 14 b show by way of example the asymmetrical variationsin the index profile in different coupling regions of the filteraccording to FIG. 13;

FIG. 15 shows the result of a measurement of the dependency of thefilter spectrum on the temperature;

FIG. 16 shows a diagram of a WDM optical transmission system; and

FIG. 17 shows an amplifier of the transmission system according to FIG.16, comprising the filter according to the invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

With reference to FIG. 3 a, 1 denotes a fibre optical filter of theMach-Zehnder type. The filter 1 includes a dual-mode optical fibre 2which has a length preferably of between 1 mm and 100 mm and comprises acore 3 and a cladding 4, both having the same longitudinal axis 5. Thefibre 2 also has a superficial protective coating 6 consisting ofpolymer material. The coating 6 is partially removed during the processfor formation of the filter 1 (as shown in the figure) and if necessarymay be reapplied at a later stage.

The core 3 has a radius r₁ and a refractive index n₁ and is composed ofsilica (SiO₂) doped with one or more elements which have the effect ofraising the refractive index, such as, for example, germanium (Ge) andphosphorus (P). As shown in FIG. 3 b, the cladding 4 comprises an innerregion 4 a and an outer region 4 b, both of which are annular in crosssection. The inner region 4 a borders with the core 3 (and therefore hasan internal radius equal to r₁), has a refractive index n₂ and has anexternal radius r₂ equal to k·r₁, where k is a suitable coefficient,preferably between 2 and 6. Moreover, r₁ is preferably between 2.5 and6.5 μm. The outer region 4 b has a refractive index n₃ and an externalradius r₃ preferably equal to 62.5 μm.

Preferably, the difference n₁−n₂ between the refractive indices of thecore 3 and the inner region 4 a lies between 3.4·10⁻³ and 1.5·10⁻².Moreover, the difference n₂−n₃ between the refractive indices of theinner region 4 a and the outer region 4 b is preferably between +1·10⁻³and −2·10⁻³. More preferably, n₂ is substantially equal to n₃ and thefibre 2 therefore has a refractive index profile substantially in theform of a step (step index), as shown in FIG. 4.

As is known, in the case of a fibre with a refractive step index profilethe cut-off wavelength λ_(c) is determined solely by the radius r₁ ofthe core 2 and the numerical aperture NA. In the present case, theradius r₁ and the numerical aperture NA are chosen so that the fibre 2is dual-mode in the spectral region currently of greatest interest foroptical telecommunications, i.e. between 1500 nm and 1650 nm.

Since the fibre 2 must be able to communicate with single-mode fibreswith known characteristics, it may be designed with a refractive stepindex n₁−n₂ and with a radius r₁ such as to have a distribution of thefundamental mode substantially equivalent to that in the single-modefibres considered. In this way, power transfer of only the fundamentalmode is ensured between the fibre 2 and these single-mode fibres.

The inner region 4 a has a composition such that it isthermo-refractive. This composition comprises silica (Si), germanium(Ge), phosphorus (P) and fluorine (F). The Applicant has ascertainedthat, with this composition, it is possible to obtain, using the thermaldisturbance technique described below, a variation in the refractiveindex equal to at least 5·10⁻⁴. Advantageously, the variation in therefractive index thus obtained may be greater than or equal to 1·10⁻³,even more advantageously greater than or equal to 2·10⁻³.

The Applicant has found that, in order to obtain an inner region 4 awith the abovementioned refractive index value and with theabovementioned thermo-refractivity characteristics, the concentrationsof the abovementioned dopants in this region must lie within thefollowing ranges:

-   -   Ge: between 2% and 5%    -   P: between 0.5% and 2%    -   F: between 1% and 2%

Still with reference to FIG. 3 a, the fibre 2 has a first and a secondmodal coupling regions 8, 9 positioned at a distance L from each otheralong the axis 5. The modal coupling regions 8, 9 are formed by inducingthermally, using the method described below, a variation in therefractive index Δn which is asymmetrical in the inner region 4 a. Inpractice, as shown in FIG. 3 b, a portion 7 (shown shaded in grey) ofthe inner region 4 a, defining a substantially annular sector in crosssection, has a refractive index greater than that of the remainder ofthe cladding cross section.

Owing to the presence, inside an optical fibre, of a zone with a veryasymmetrical refractive index profile, it is possible to achieve strongpower coupling between the fundamental mode LP₀₁ and the asymmetricalmode LP₁₁. Each of the modal coupling regions 8, 9 therefore defines,together with the fibre 2, a modal coupler. Preferably, the variation inthe index must be of a form (in the section considered) very similar tothat of the mode LP₁₁.

The section of fibre 2 lying between the two coupling regions 8,9—denoted by 10—is referred to below as the “phase shift region” sinceit defines the region in which the modes LP₀₁ and LP₁₁ undergo a mutualphase shift Δφ which is a function of the wavelength. The filter 1therefore defines two coupling regions 8, 9 and a phase-shift region 10lying between them.

Furthermore, as shown in the schematic illustration in FIG. 3 c, thefilter 1 comprises a first and a second fibres 11, 12 of the standardsingle-mode (SM) type, which are connected by means of respective jointsat the opposite ends of the fibre 2, so as to allow a substantiallyloss-free coupling with the single-mode transmission fibres of thesystem in which the filter 1 is placed. The fibres 11, 12 aresingle-mode in a spectral band lying between 1500 nm and 1650 nm. Thefilters 11, 12 define, respectively, an input for single-mode signalsS_(in) to be filtered and an output for the filtered single-mode signalsS_(out). The fibres 11, 12 have geometric characteristics such that theyhave a profile of the fundamental mode the same as that of the fibre 2,so as to minimise the coupling losses therewith and perform modalfiltering in order to eliminate the mode LP₁₁. The fibres 11, 12preferably have a numerical aperture NA of between 0.1 and 0.2 and anexternal radius (of the cladding) equal to about 62.5 μm.

The operation of the filter 1 is described below. When a single-modeoptical input signal S_(in) reaches, via the first single-mode fibre 11,the first coupling region 8, a transfer of power occurs from the modeLP₀₁ to the mode LP₁₁ in a quantity dependent on the wavelength.Subsequently, the modes LP₀₁ and LP₁₁ are propagated in the phase shiftregion 10, at the end of which they have a phase difference Δφ expressedby the following relation:

$\begin{matrix}{{\Delta\phi} = {{\frac{2\pi}{\lambda} \cdot \Delta}\;{n_{eff} \cdot L}}} & (3)\end{matrix}$

where Δn_(eff) is the difference between the effective refractiveindices of the mode LP₀₁ and the mode LP₁₁ and λ is the wavelength.

This phase difference is due to the different optical paths followed bythe modes LP₀₁ and LP₁₁ owing to their different effective refractiveindices n_(eff) and, therefore, to their different speeds of propagationwithin the phase shift region 10. When the two modes LP₀₁ and LP₁₁ reachthe second modal coupling region 9, they are combined again, interferingconstructively or destructively depending on the wavelength considered.The outgoing signal from the second modal coupling region is furtherfiltered upon entering into the second single-mode fibre 12, withelimination of the mode LP₁₁. A single-mode signal S_(out) with aspectral form depending on the spectral response of the filter 1 istherefore output from the fibre 12.

The process for manufacturing the filter 1 is described below.

The optical fibre 2 is made using the technique of modified chemicalvapour deposition (MCVD). In this process, in order to obtain thedesired composition in the thermo-refractive inner region 4 a, then, inaddition to the oxygen and silicon tetrachloride (SiCl₄) which aretypically used in this process, germanium tetrachloride (GeCl₄),phosphorus oxychloride (POCl₃) and one of the following compounds offluorine: Freon (CCl₂F₂), sulphur hexafluoride (SF₆) and silicontetrafluoride (SiF₄), are also introduced into the deposition tube.

The coupling regions 8, 9 are then formed on the optical fibre 2. TheApplicant has found that the coupling regions 8, 9 may be formed byapplying, to the fibre 2, an asymmetrical thermal disturbance able toproduce the desired variation in the refractive index profile in theinner region 4 a of the cladding 4. The Applicant has also found thatthis thermal disturbance may be produced by means of an electric arc.

With reference to FIG. 5, 13 denotes an apparatus for forming thecoupling regions, comprising a fusion jointer 14 of known type, forexample a Fujikura model FSM-20CSII fusion jointer for optical fibres,and a fibre moving device 15, able to perform micrometric displacementsof the fibre 2 parallel to its axis.

In order to induce an asymmetrical thermal disturbance, the fibre 2 ispositioned between the electrodes of the jointer 14, indicated by 16,17, as shown in FIG. 6.

The jointer 14 is then activated so as to produce an electric arc 18which causes sudden heating of the fibre 2 and, after the discharge,subsequent rapid cooling thereof. Since the position of the fibre 2 isnever perfectly symmetrical with respect to the electrodes 16, 17, theelectric arc 18 is usually formed only on one side of the section of thefibre 2, as shown in the Figure. There is therefore a temperaturedistribution inside the fibre 2 such as to cause an asymmetricalvariation in the refractive index Δn. This behaviour of the electric arc18 may be observed, for example, by positioning a videocamera (notshown) close to the electrodes 16, 17 of the jointer 14.

In order to form the other coupling region, the fibre 2 must bedisplaced parallel to its axis 4 by means of the fibre moving device 15so as to arrange, between the electrodes 16, 17, a different portion offibre 2, the distance of which from the previously treated portion isexactly equal to L, and apply again the electric arc to this portion.

The Applicant has manufactured, in order to carry out some experimentalmeasurements described below, a fibre 2 with the followingcharacteristics:

-   -   r₁ equal to 4.7 μm;    -   k(=r₂/r₁) equal to 4.3;    -   n₂=n₃;    -   numerical aperture NA equal to 0.15;    -   cut-off wavelength λ_(c) equal to 1630 nm;    -   inner region 4 a comprising (by scanning electron microscope        (SEM) analysis): 95.2% silica (Si), 4% germanium (Ge), 0.8%        phosphorus (P). The percentage of fluorine (F), which cannot be        determined using the SEM technique, was estimated at about 1.3%        using the teaching of K. Abe, European Conference on Optical        Fiber Communication, Paris, 1996, Presentation II.4, taking into        account that this concentration allows to achieve the same        refraction index value in the inner region 4 a and in the outer        region 4 b.

From this fibre, a filter 1 with the following additionalcharacteristics was produced:

-   -   distance L between the coupling regions 8 and 9: 30 mm;    -   numerical aperture of the fibres 11, 12: 0.12;    -   cut-off length of the fibres 11, 12: 1200 nm;    -   external radius of the cladding of the fibres 11, 12: 62.5 μm.

FIGS. 7 a and 7 b show, respectively, the refractive index profile ofthe fibre thus obtained in the thermally disturbed section and in asection which is not thermally disturbed. From FIG. 7 a it can be seenthat the variation Δn in the refractive index profile is asymmetrical inthe section of the fibre 2 and has a maximum value of about 2·10⁻³.

The characteristics of the coupling regions 8, 9 are determined by thepower and the duration of the electric arc 18. The Applicant has notedthat it is not possible to establish precisely, on the basis of theparameters of the electric arc 18, the amount of the variation in theindex profile and, therefore, the coupling factor. In order to verifythe coupling properties of the regions 8, 9, it is possible to performmonitoring, during the writing process, of the extinction ratio of thefilter (which is correlated to the coupling factor), by means of aspectral analysis. FIG. 8 shows an apparatus 24 which can be used formonitoring the coupling characteristics of the filter during theformation of the coupling regions 8, 9. The apparatus 24 comprises awhite light source 25 able to supply wide-spectrum electromagneticradiation to the fibre 2, a spectrum analyser 26 able to analyse thespectrum of the light leaving the fibre 2, and a processing unit 27connected to the spectrum analyser 26 for processing informationsupplied by the said analyser.

FIG. 9 shows the transmission spectrum, obtained by means of theapparatus 24, of a filter 1 with the characteristics described above.From this figure it is possible to note that the spectral form of afilter produced in accordance with the invention is that typical of theinterferential filters of the Mach-Zehnder type, i.e. is periodic with aperiodicity depending on the wavelength considered. The extinction ratiothus obtained (namely the difference between the minimum and maximumtransmissivity of the filter expressed in dB) is equal to about 1.2 dBand the insertion losses are equal to about 0.4 dB. The Applicant hasalso noted that, by optimising the process parameters, it is possible toobtain an extinction ratio greater than 2.5 dB.

On the basis of the desired spectral response, the optical filteringdevice according to the present invention may comprise, in a manner notshown, several filtering stages arranged in cascade. In other words,this device may comprise a plurality of filters 1 which are connected inseries so as to have a spectral response determined by the combinationof the responses of the various filters. As known from the text “FiberOptic Networks”, Prentice Hall, P. E. Green, 1993, page 123, in order todesign a Mach-Zehnder filter with a desired spectral behaviour, it isnecessary to know the dispersion characteristic of the modes which arepropagated along the fibre, namely the value Δβ (λ) of the differencebetween the propagation constants of the interfering modes. The modaldispersion characteristic may be obtained by means of regression or“fitting” of the spectral response (for example that shown in FIG. 9) ofa test filter of known length. From this dispersion characteristic, itis possible, by means of digital simulation, to determine the parametersof the interferometer, in particular the distance L (or the distances Lbetween the coupling points, in the case of several interferometersarranged in cascade) and the values of the coupling coefficients, whichare required in order to produce the filter with the desired spectralresponse.

The efficiency with which the asymmetrical variation in the indexprofile obtained using the technique according to the invention inducescoupling in the higher asymmetrical mode LP₁₁ may be verified by meansof a suitable experimental test. For this purpose it is possible to usea measuring apparatus such as that shown in FIG. 10 and indicatedtherein by 19.

The measuring apparatus 19 comprises a laser source 20 able to supply toone of the ends of the fibre 2 a laser beam at the wavelength of 1550 nmand a infrared videocamera 21 positioned so as to be able to detect thelight emitted from the fibre. In particular, the camera 21 is able todetect the intensity profile of the electromagnetic field (known as“near field”) emitted from the fibre 2. The measuring apparatus 19 alsocomprises a processing unit 22 connected to the camera 21 so as toreceive from it a digital signal correlated with the optical signaldetected.

The intensity profile of the electromagnetic field detected by thecamera 21 is formed by the superimposition of the modes which arepropagated in the fibre 2 and, in mathematical terms, is defined by thesquare of the linear combination of these modes. Each mode also has,associated with it, a multiplication coefficient which determines itsamplitude and, therefore, its weight within the linear combination. Inorder to derive these coefficients it is possible to perform a linearregression (or fitting) operation on the result of the experimentalmeasurement. In practice, based on the distribution of the fibre modes(LP₀₁, LP₁₁, etc.), these modes are combined so as to obtain theintensity of the resultant field which best approximates that measured.

The Applicant carried out a test using a fibre 2 having thecharacteristics described above and provided with the coupling regions 8and 9. FIGS. 11 a and 11 b show the linear regression (fitting)coefficients, the first for the even modes (of the type LP_(0m)) and thesecond for the odd modes (of the type LP_(1m)) obtained from theanalysis of the fibre 2. These graphs confirm that the only modesinvolved in the coupling are the modes LP₀₁ and LP₁₁. In the case inquestion, the values of the coefficients associated with the modes LP₀₁and LP₁₁ are equal to 0.79 and 0.21 respectively. This measurementtherefore confirms that the coupling induced by means of theasymmetrical variation in the index profile produces a high modalselectivity, resulting in a practically negligible contribution of modesother than the modes LP₀₁ and LP₁₁.

The Applicant has also noted that the coupling factor, defined as beingthe ratio between the power transferred to the mode LP₁₁ and thatremaining in the mode LP₀₁, increases with the intensity (in other wordswith the amperage) of the electric arc. However, the Applicant has alsonoted that if this intensity is too high, a geometric deformation of thefibre is induced, in addition to a variation Δn in the refractive indexin the fibre. This deformation causes power losses which involve anincrease in the insertion losses of the filter and, therefore, adeterioration in the performance of the said filter. It is thereforenecessary to achieve a compromise between the desired coupling factor(and therefore the desired extinction) and the resultant insertionlosses. The Applicant has ascertained that the electric arc must have aduration preferably of less than 400 ms, more preferably less than 300ms, and a current intensity preferably between 8 and 14 mA, morepreferably between 10 and 11 mA. More preferably, instead of a singlearc, a sequence of arcs with the abovementioned characteristics may beapplied.

The Applicant also noted that, since the index profile variation whichcauses coupling does not have a circular symmetry, the coupling factorvaries in accordance with polarisation of the light. The operation ofthe filter 1 therefore depends on the polarisation of the incominglight. This dependency is measured by evaluating, for each wavelength,the maximum variation which exists in the attenuation spectrum of thefilter with variation in the polarisation (PDL, Polarisation DependentLoss). FIG. 12 shows the PDL measured, using a known technique, on afilter 1 which has the characteristics described above. The mean valueof the measured PDL is about 0.4 dB, for a filter with an extinctionratio of about 2.6 dB.

The Applicant notes that this dependency on the polarisation may bedisadvantageous when the filter 1 is used in an amplification stage.

FIG. 13 shows schematically a variation of the filter according to theinvention—denoted by 1′—able to reduce significantly the abovementionedproblem. The filter 1′ differs from the filter 1 in that two furthercoupling regions 8′ and 9′ are present, preferably at a distance fromeach other equal to L. The coupling regions 8′ and 9′ differ from thecoupling regions 8, 9 in that the former have an asymmetrical variationin the index profile which is perpendicular to that of the latter. Inparticular the coupling regions 8′ and 9′ have, in cross section, anannular sector 7′ which is rotated through a right angle (90°) withrespect to the annular sector 7 of the coupling regions 8 and 9. FIGS.14 a and 14 b show, by way of example, the asymmetrical variations inthe index profile in the coupling regions 8 and 8′, respectively(similar to those present in the regions 9 and 9′, respectively). Themutual distance between the coupling regions 8 and 8′ and between thecoupling regions 9 and 9′ is preferably the same, for example 100 μm.Since this distance is very small, the undesirable effects of modulationof the signal due to the presence of the additional coupling regions 8′and 9′ is negligible.

As before, it is possible to produce an optical filtering devicecomprising, in a manner not shown, a plurality of filters 1′ connectedin series.

The Applicant has also noted that the operation of the filter 1 dependson the operating temperature. In particular, with a variation in thetemperature, the peaks in the spectral response of the filter 1 aredisplaced in terms of wavelength. In order to verify the sensitivity totemperature of the filter according to the present invention, a filter 1with the characteristics indicated above was positioned in acontrolled-temperature chamber, in which the temperature was varied (forexample with a ramp-like variation) so as to cause the displacement, inwavelength, of its resonance peaks. FIG. 15 shows the results of thismeasurement. In particular, the points measured and a regression(fitting) line for a filter with a distance L of 20 mm are shown. It wasfound that, for each millimetre of length of the filter, the position ofthe peak in the spectrum varies by about 0.0016 nm for each degreecentigrade of variation in the temperature. The Applicant notes thatthis dependency is substantially equivalent to that demonstrated byother interferential filters of the Mach-Zehnder type.

The filter according to the present invention may be advantageously usedin a long-distance WDM (Wavelength Division Multiplexing)telecommunications system, for example an undersea telecommunicationssystem.

As shown in FIG. 16, an optical telecommunications system typicallycomprises a transmission station 32, a receiving station 33 and anoptical communications line 34 connecting the transmission station 32and receiving station 33. The transmission station 32 comprises aplurality of optical transmitters 35, each of which is able to transmitan optical signal at a respective wavelength. Each optical transmitter35 may, for example, comprise a source of the laser type and awavelength converter able to receive the signal generated by the laserand transmit a signal at a predefined wavelength. A wavelengthmultiplexer 36 is connected on its input side to the transmitters 35 soas to receive the plurality of signals transmitted and has a singleoutput connected to the communication line 34 in order to transmit thewavelength multiplexed signals on the line. The transmission station 32may also comprise an optical power amplifier 37, which is connected tothe output of the multiplexer 36, so as to impart to the signalstransmitted the necessary power for transmission along the line 34.

The receiving station 32 comprises a wavelength demultiplexer 38connected at its input to the line 34 so as to receive the signalstransmitted and has a plurality of outputs into which the variouswavelengths transmitted are divided. The receiving station 32 alsocomprises a plurality of optical receivers 39, each connected to arespective output of the demultiplexer 38 in order to receive a signalat a respective wavelength. Each receiver 39 may comprise a wavelengthconverter to convert the wavelength of the signal into a wavelengthsuitable for reception of the signal by a photo-detector connectedoptically to the said converter. The receiving station 32 may alsocomprise a pre-amplifier 40 arranged upstream of the demultiplexer 38 soas to impart to the signals transmitted the power necessary for correctreceiving thereof.

The communication line 34 comprises many sections of optical fibre 41(preferably single-mode optical fibre) and a plurality of lineamplifiers 42 located at a distance from each other (for example ahundred kilometres or so) and designed to amplify the signals to a powerlevel suitable for transmission to the next optical fibre section.

As shown schematically in FIG. 17, at least one of the amplifiers of thetransmission system (i.e. the power amplifier 37, the pre-amplifier andthe line amplifiers 42), denoted here by 45, is a two-stage amplifier,i.e. it comprises a first and a second active fibres 46, 47 foramplification of the signals, connected in series. As shown, the filter1 according to the invention may be positioned between the twoamplification stages so as to perform equalization of the signals.Alternatively, the filter may be positioned downstream of the twostages.

Lastly, the Applicant has found that the filter according to theinvention may be effectively used also as a temperature or deformationsensor since its spectral response is sensitive to variations intemperature and length in accordance with known laws. In particular, bydetecting the displacement of predefined points in the filter spectrumit is possible to determine the variation in the parameter measured.

During operation as a temperature sensor, the sensor may be used inorder to measure the absolute temperature present in a givenenvironment, after being calibrated to a predefined temperature. In asimilar manner, it may be used to measure variations in temperature.

During operation as a deformation sensor, the filter 1 is applied to abody liable to undergo deformation. The variation in the spectralresponse of the filter 1 following deformation of the body provides ameasurement of the said deformation.

1. An optical fiber filter, comprising: an optical fiber; a pair offirst coupling regions; and a phase-shift region; wherein the opticalfiber comprises: a core; and a cladding; wherein an optical signal canpass through the optical fiber, wherein the first coupling regions areformed in the optical fiber, at a predefined mutual distance, forproducing power transfer between a first propagation mode of the opticalsignal and a second propagation mode of the optical signal, wherein thephase-shift region is defined by a section of the optical fiber,disposed between the first coupling regions, for producing a phase shiftbetween the first and second propagation modes, and wherein refractiveindex profiles of the optical fiber in each of the first couplingregions are asymmetrical due to the presence of a cladding zone with arefractive index higher than a refractive index of the remainder of thecladding.
 2. The filter of claim 1, wherein, in each of the firstcoupling regions, the cladding comprises, in cross-section, an annularsector, and wherein, in each annular sector, a refractive index of theannular sector is greater than a refractive index of a remainder of thecladding.
 3. The filter of claim 2, wherein the cladding comprises, incross-section: an inner annular region; and an outer annular region;wherein the inner annular region is adjacent to the core, and whereinthe inner annular region comprises the annular sectors.
 4. The filter ofclaim 2, wherein the annular sectors are disposed in asubstantially-similar angular position.
 5. The filter of claim 2,further comprising: a pair of second coupling regions formed in theoptical fiber; wherein the annular sectors are disposed in a firstsubstantially-similar angular position, wherein each of the secondcoupling regions comprises, in cross-section, a further annular sectorcomprising a refractive index greater than the refractive index of theremainder of the cladding, wherein the further annular sectors aredisposed in a second substantially-similar angular position, and whereinthe first substantially-similar angular position is different from thesecond substantially-similar angular position.
 6. The filter of claim 5,wherein each of the second coupling regions is formed in a vicinity of arespective first coupling region.
 7. The filter of claim 2, wherein adifference between the refractive index of each of the annular sectorsand the refractive index of the remainder of the cladding is greaterthan or equal to 5×10⁻⁴.
 8. The filter of claim 1, wherein the claddingcomprises, in cross-section: an inner annular region; and an outerannular region; wherein the inner annular region is adjacent to thecore.
 9. The filter of claim 1, wherein, outside of the first couplingregions, the optical fiber comprises, in cross-section, a refractiveindex profile of a step-index type.
 10. The filter of claim 1, whereinthe optical signal comprises a wavelength comprised in a predefinedtransmission wavelength band, and wherein the optical fiber is dual-modein the predefined transmission wavelength band.
 11. The filter of claim8 or 3, wherein the inner annular region comprises silica, and whereinthe inner annular region comprises oxides of at least one of germanium,phosphorus, and fluorine.
 12. An optical fiber filtering device,comprising: a plurality of optical fiber filters; wherein the opticalfiber filters are connected In series, wherein each optical fiber filtercomprises: an optical fiber; a pair of coupling regions; and aphase-shift region; wherein the optical fiber comprises: a core; and acladding; wherein an optical signal can pass through the optical fiber,wherein the coupling regions are formed in the optical fiber, at apredefined mutual distance, for producing power transfer between a firstpropagation mode of the optical signal and a second propagation mode ofthe optical signal, wherein the phase-shift region is defined by asection of the optical fiber, disposed between the coupling regions, forproducing a phase shift between the first and second propagation modes,and wherein refractive index profiles of the optical fiber in each ofthe coupling regions are asymmetrical due to the presence of a claddingzone with a refractive index higher than a refractive index of theremainder of the cladding.
 13. An optical amplifier, comprising: anoptical fiber filter; and at least one optical amplification stage;wherein the optical fiber filter is disposed in series with the at leastone optical amplification stage, wherein the optical fiber filtercomprises: an optical fiber; a pair of coupling regions; and aphase-shift region; wherein the optical fiber comprises: a core; and acladding; wherein an optical signal can pass through the optical fiber,wherein the coupling regions are formed in the optical fiber, at apredefined mutual distance, for producing power transfer between a firstpropagation mode of the optical signal and a second propagation mode ofthe optical signal, wherein the phase-shift region is defined by asection of the optical fiber, disposed between the coupling regions, forproducing a phase shift between the first and second propagation modes,and wherein refractive index profiles of the optical fiber in each ofthe coupling regions are asymmetrical due to the presence of a claddingzone with a refractive index higher than a refractive index of theremainder of the cladding.
 14. An optical telecommunications system,comprising: at least one optical transmitter; at least one opticalreceiver; an optical transmission line connecting the at least oneoptical transmitter to the at least one optical receiver; and at leastone optical amplifier: wherein the at least one optical amplifier isdisposed along the optical transmission line, wherein the at least oneoptical amplifier comprises: an optical fiber filter; and at least oneoptical amplification stage; wherein the optical fiber filter isdisposed in series with the at least one optical amplification stage,wherein the optical fiber filter comprises: an optical fiber; a pair ofcoupling regions; and a phase-shift region; wherein the optical fibercomprises: a core; and a cladding; wherein an optical signal can passthrough the optical fiber, wherein the coupling regions are formed inthe optical fiber, at a predefined mutual distance, for producing powertransfer between a first propagation mode of the optical signal and asecond propagation mode of the optical signal, wherein the phase-shiftregion is defined by a section of the optical fiber, disposed betweenthe coupling regions, for producing a phase shift between the first andsecond propagation modes, and wherein refractive index profiles of theoptical fiber in each of the coupling regions are asymmetrical due tothe presence of a cladding zone with a refractive index higher than arefractive index of the remainder of the cladding.
 15. An optical fibermodal coupler, comprising: an optical fiber; and a coupling region;wherein the optical fiber comprises: a core; and a cladding; wherein anoptical signal can pass through the optical fiber, wherein the couplingregion is formed in the optical fiber to produce power transfer betweena first propagation mode of the optical signal and a second propagationmode of the optical signal, and wherein a refractive index profile ofthe optical fiber in the coupling region is asymmetrical due to thepresence of a claddin zone with a refractive index higher than arefractive index of the remainder of the cladding.